Subgroups — #65787

Order 320.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#713	C₂	1	1	yes	maximal, characteristic
2	C₂×#700	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#4003	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×Q₈	C₄	1	1	yes	characteristic
6	#713	—	2	2	no	—
7	#713	—	2	2	no	—
8	C₂×D₂₀	C₄	1	2	yes	—
9	C₂×D₂₀	C₄	1	1	yes	characteristic
10	C₂×C₄×D₅	C₂²	1	2	yes	—
11	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
12	C₂×#170	C₂²	1	2	yes	—
13	C₂×#170	C₂²	1	2	yes	—
14	#700	—	2	2	no	—
15	D₅⋊C₈	—	2	2	no	—
16	#408	—	5	1	no	maximal, Sylow
17	C₂²×D₅	C₂×C₄	1	2	yes	—
18	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
19	C₅×Q₈	—	2	2	no	—
20	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₅	C₂³	1	1	yes	characteristic
23	D₂₀	—	2	4	no	—
24	D₂₀	—	2	2	no	—
25	C₄×D₅	—	2	2	no	—
26	C₄×D₅	—	2	2	no	—
27	C₄×D₅	D₄	1	2	yes	—
28	C₄×D₅	D₄	1	2	yes	—
29	#170	—	2	2	no	—
30	#170	—	2	2	no	—
31	C₂×#41	—	5	1	no	—
32	C₂×M₁₆	—	5	1	no	—
33	C₂²×C₈	—	5	1	no	—
34	C₂²⋊C₈	—	5	4	no	—
35	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₅	—	2	2	no	—
37	C₂×D₅	—	2	2	no	—
38	C₂×D₅	—	2	4	no	—
39	C₂×D₅	C₂²⋊C₄	1	2	yes	—
40	C₄×C₅	—	2	2	no	—
41	C₄×C₅	C₂²⋊C₄	1	2	yes	—
42	Dic₅	C₂×D₄	1	1	yes	characteristic
43	Dic₅	C₂×D₄	1	1	yes	characteristic
44	#41	—	10	2	no	—
45	#41	—	10	2	no	—
46	C₂×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₂×D₄	—	5	2	no	—
48	C₂×D₄	—	5	1	no	—
49	C₂²×C₄	—	5	1	no	—
50	C₂²×C₄	—	5	2	no	—
51	M₁₆	—	10	2	no	—
52	C₂×C₈	—	5	2	no	—
53	C₂×C₈	—	5	2	no	—
54	C₂×C₈	—	10	2	no	—
55	C₂×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₅	#131	1	2	yes	—
57	D₅	—	2	2	no	—
58	D₅	—	4	2	no	—
59	C₂³	—	5	1	no	—
60	C₂³	—	5	2	no	—
61	Q₈	—	2	2	no	—
62	D₄	—	10	4	no	—
63	D₄	—	10	2	no	—
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	5	2	no	—
66	C₂×C₄	—	5	2	no	—
67	C₂×C₄	—	10	2	no	—
68	C₂×C₄	—	10	2	no	—
69	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
70	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
71	C₈	—	10	2	no	—
72	C₈	—	10	2	no	—
73	C₅	#408	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	5	2	no	—
75	C₂²	—	10	2	no	—
76	C₂²	—	10	4	no	—
77	C₂²	—	10	2	no	—
78	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	5	1	no	—
81	C₄	—	5	1	no	—
82	C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
83	C₂	—	10	2	no	—
84	C₂	—	20	2	no	—
85	C₂	#4133	1	2	yes	—
86	C₂	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₂×C₅×Q₈	C₄
prime radical	C₂×#713	C₂
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#408
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#700	C₂	semidirect
3. C₂×D₅⋊C₈	C₂	semidirect
4. #4003	C₂	semidirect
73. C₅	#408	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#713	2
C₂×#700	2
C₂×D₅⋊C₈	2
#4003	2
#408	5